On Generalized Fractional Spin, Fractional Angular Momentum, Fractional Momentum Operators in Quantum Mechanics
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Rami Ahmad El-Nabulsi
On Generalized Fractional Spin, Fractional Angular Momentum, Fractional Momentum Operators in Quantum Mechanics
Received: 18 May 2020 / Accepted: 3 July 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract In this study, we have extended the idea of fractional spin introduced recently in literature based on two orders fractional derivative operator. Generalizations of the fractional spin, the fractional angular momentum and the fractional momentum operators were obtained. The theory is characterized by a noncommutativity between the generalized fractional angular momentum and the fractional Hamiltonian. We have derived the corresponding fractional Schrödinger equation and we have discussed its implications on the problems of a free particle and a particle moving in an infinite well potential. Enhancements of their corresponding energies levels and ground energies were observed which are in agreement with phenomenological theories such as noncommutative quantum mechanics.
1 Introduction Fractional calculus has been proved recently to a powerful mathematical tool to explain dynamical systems and physical processes characterized by long-range memory and fractal aspects [1]. Its implications in several branches of sciences and engineering ranging from turbulent flow, to polymer physics, kinetic theories, material sciences, finance and economics, astrophysics and cosmology, among other fields are highly appreciated [2– 39]. In quantum mechanics and quantum field theories, several applications with motivating results were obtained in literature. Some of these include the fractional uncertainty relation [40–43], the quantization of fractional derivatives operators [44], the fractional wave theories [45–50], the fractional path integral [29,51– 54], the fractional spacetime approaches [55–57], the fractional quantum field theory at positive temperature [58,59], etc. It should be stressed that a connection between fractional calculus and fractals is well-known in literature [60–63]. Since fractional calculus allows us to describe quantum field theory in the language of fractional differential equations, it was revealed recently that fractional derivatives and integrals operators provide a practical device to deal more truthfully with quantum dynamics characterized by multiple scales generated in the deep ultraviolet regime of quantum field theory [64–70]. The instability of quantum vacuum fluctuations produced on long-time scales lead to self-organized criticality and this represents one of the main reasons to use fractional calculus in quantum field theory. In quantum mechanics and particle physics, several new insights were obtained in literature, e.g. the emergence of fractional spin based on the fractional extended translation-rotation-like property for particles described with the fractional Schrödinger equation [71], the construction of fractional Zeeman’ effect to reproduce the baryon spectrum accurately [72], the formulation of fractional symmetric rigid operator used
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